The name Ashkenaz in the Bible
Since Genesis 10 deals largely with peoples, it's safe to assume that to the authors of it, Ashkenaz represented a certain nation. Which nation that was is unclear, but in the Middle Ages the name Ashkenaz became applied to the Jews of Western Europe, the Ashkenazi, initially centered in northern Italy. Their (middle-)eastern counterpart eventually moved to Spain (which is obviously more western than Italy), and became known as the Sephardi Jews.
Etymology of the name Ashkenaz
The name Ashkenaz, like other names from the first few chapters of Genesis, apparently stem from deep antiquity, and we're not sure what they are supposed to mean, or even from what language they stem. But since they were written down to serve a Hebrew audience, it may have been spelled in such as way that it came to mean something, and for some reason.
Neither NOBSE Study Bible Name List, nor BDB Theological Dictionary offers any explanation of this name but Jones' Dictionary of Old Testament Proper Names proposes the following: Jones suggests that the name Ashkenaz can be seen as to consist of three parts:
(1) The noun אש ('esh) meaning fire:
2) The Hebrew comparative particle כ (The particle ke) meaning as, like.
3) The verb נזה (naza), meaning to sprinkle:
Whether intentional or not, for a Hebrew audience the name Ashkenaz would have sounded as Fire Like Sprinkles. Jones' Dictionary of Old Testament Proper Names renders this name as So Fire Is Scattered.